Fuzzy Stability of Quadratic Functional Equations

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doi:10.1155/2010/412160

Research Article

Fuzzy Stability of Quadratic Functional Equations

Jung Rye Lee,

1

_{Sun-Young Jang,}

2

_{Choonkil Park,}

3

and Dong Yun Shin

4

1_{Department of Mathematics, Daejin University, Kyeonggi 487-711, Republic of Korea} 2_{Department of Mathematics, University of Ulsan, Ulsan 680-749, Republic of Korea}

3_{Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791,}

Republic of Korea

4_{Department of Mathematics, University of Seoul, Seoul 130-743, Republic of Korea}

Correspondence should be addressed to Dong Yun Shin,[email protected]

Received 10 February 2010; Accepted 11 April 2010

Academic Editor: T. Bhaskar

Copyrightq2010 Jung Rye Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated by Moslehian et al. In this paper, we prove the generalized Hyers-Ulam stability of the following quadratic functional equationsfxyfx−y 2fx2fyandfaxbyfax−by 2a2_fx₂_b2_{fy a, b}_∈ R\{0}, a /±1in fuzzy Banach spaces.

1. Introduction and Preliminaries

Katsaras1defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view2–4. In particular, Bag and Samanta5, following Cheng and Mordeson6, gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Mich´alek type7. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces8.

We use the definition of fuzzy normed spaces given in5,9,10to investigate a fuzzy version of the generalized Hyers-Ulam stability for the quadratic functional equations

fxyfx−y2fx 2fy, 1.1 faxbyfax−by2a2_f_x _2b2_f_y _1.2

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Definition 1.1see5,9,10. LetX be a real vector space. A functionN :X×R → 0,1is called afuzzy normonXif, for allx, y∈Xand alls, t∈R,

N1Nx, t 0 fort≤0,

N2x0 if and only ifNx, t 1 for allt >0,

N3Ncx, t Nx, t/|c|ifc /0,

N4Nxy, st≥min{Nx, s, Ny, t},

N5Nx,·is a nondecreasing function ofRand limt→ ∞Nx, t 1,

N6forx /0,Nx,·is continuous onR.

The pairX, Nis called afuzzy normed vector space.

The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in9,10.

Definition 1.2see5,9,10. LetX, Nbe a fuzzy normed vector space. A sequence{xn}in Xis said tobe convergentorconvergesif there exists anx∈Xsuch that lim_n_{→ ∞}Nx_n−x, t 1 for allt > 0. In this case,xis called thelimitof the sequence {xn}and we denote it by N-limn→ ∞xnx.

Definition 1.3see5,9,10. LetX, Nbe a fuzzy normed vector space. A sequence{x_n}in X is calledCauchyif for each ε >0 and eacht > 0 there exists ann0 ∈ Nsuch that, for all n≥n0and allp >0, we haveNxnp−xn, t>1−ε.

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to becomplete and the fuzzy normed vector space is called afuzzy Banach space.

We say that a mappingf :X → Y between fuzzy normed vector spacesX and Y is continuous at a pointx0 ∈X if, for each sequence{xn}converging tox0inX, the sequence {fxn}converges tofx0. Iff :X → Yis continuous at eachx∈X, thenf:X → Yis said to becontinuousonXsee8.

The stability problem of functional equations is originated from a question of Ulam

11concerning the stability of group homomorphisms. Hyers12gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki 13 for additive mappings and by Th. M. Rassias 14 for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias14has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by G˘avrut¸a 15by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias’ approach.

A square norm on an inner product space satisfies the parallelogram equality

_x_y2_x₋_y2

2x22y2. 1.3

The functional equation

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is called aquadratic functional equation. In particular, every solution of the quadratic functional equation is said to be aquadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof16for mappingsf :X → Y, where X is a normed space and Y is a Banach space. Cholewa17 noticed that the theorem of Skof is still true if the relevant domainX is replaced by an Abelian group. In18, Czerwik proved the generalized Hyers-Ulam stability of the quadratic functional equation. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappingssee19–31.

This paper is organized as follows. InSection 2, we prove the generalized Hyers-Ulam stability of the quadratic functional equation1.1in fuzzy Banach spaces. InSection 3, we prove the generalized Hyers-Ulam stability of the quadratic functional equation1.2in fuzzy Banach spaces.

Throughout this paper, assume that X is a vector space and thatY, N is a fuzzy Banach space. Leta, bbe nonzero real numbers witha / ±1.

2. Generalized Hyers-Ulam Stability of the Quadratic

Functional Equation

1.1

In this section, we prove the generalized Hyers-Ulam stability of the quadratic functional equation1.1in fuzzy Banach spaces.

Theorem 2.1. Letϕ:X2 _→ _0,_∞_{be a function such that}

ϕx, y: ∞

n0

4−nϕ2nx,2ny<∞ 2.1

for allx, y∈X. Letf:X → Ybe a mapping withf0 0such that

lim t→ ∞N

fxyfx−y−2fx−2fy, tϕx, y1 _2.2

uniformly onX×X. ThenQx : N-lim_n_{→ ∞}f2nx/4nexists for eachx ∈ X and defines a quadratic mappingQ:X → Y such that if for someδ >0, α >0

Nfxyfx−y−2fx−2fy, δϕx, y≥α 2.3

for allx, y∈X, then

N

fx−Qx,δ 4ϕx, x

≥α 2.4

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Furthermore, the quadratic mappingQ:X → Yis a unique mapping such that

lim t→ ∞N

fx−Qx, tϕx, x1 _2.5

uniformly onX.

Proof. For a givenε >0, by2.2, we can find somet0>0 such that

Nfxyfx−y−2fx−2fy, tϕx, y≥1−ε 2.6

for allt≥t0. By induction onn, we show that

N 4nfx−f2nx, t n−1

k0

4n−k−1ϕ2kx,2kx≥1−ε 2.7

for allt≥t0, allx∈X,and alln∈N. Lettingyxin2.6, we get

N4fx−f2x, tϕx, x≥1−ε 2.8

for allx∈Xand allt≥t0. So we get2.7forn1. Assume that2.7holds forn∈N. Then

N 4n1fx−f2n1x, t n

k0

4n−kϕ2kx,2kx

≥min

N 4n1fx−4f2nx, t0 n−1

k0

4n−kϕ2kx,2kx,

N4f2nx−f2n1x, t0ϕ2nx,2nx

≥min{1−ε,1−ε}1−ε.

2.9

This completes the induction argument. Lettingtt0and replacingnandxbypand 2nxin

2.7, respectively, we get

N f2nx 4n −

f2npx 4np ,

t0 4np

p−1

k0

4p−k−1ϕ2nkx,2nkx≥1−ε 2.10

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It follows from2.1and the equality

The first four terms on the right-hand side of the above inequality tend to 1 asn → ∞, and the fifth term is greater than

Nf2nxyf2nx−y−2f2nx−2f2ny, t0ϕ

2nx,2ny, 2.15

which is greater than or equal to 1−ε. Thus

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for allt > 0. Since NQxy Qx−y−2Qx−2Qy, t 1 for allt > 0, byN2,

for all positive integersn. Lett >0. We have

Nfx−Qx, δϕnx, x t≥min

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lim t→ ∞N

fx−Qx, 8

4−2ptθx

p₁ _2.29

uniformly onX.

Proof. Defineϕx, y:θxpypand applyTheorem 2.1to get the result.

Similarly, we can obtain the following. We will omit the proof.

Theorem 2.3. Letϕ:X2 _→ _0,_∞_{be a function such that}

ϕx, y: ∞

n1 4nϕx

2n, y 2n

<∞ 2.30

for allx, y ∈ X. Letf : X → Y be a mapping satisfying2.2andf0 0. Then Qx : N-lim_n_{→ ∞}4nfx/2nexists for eachx∈Xand defines a quadratic mappingQ:X → Y such that if for someδ >0, α >0

Nfxyfx−y−2fx−2fy, δϕx, y≥α 2.31

N

fx−Qx,δ 4ϕx, x

≥α 2.32

for allx∈X.

lim t→ ∞N

fx−Qx, tϕx, x1 _2.33

uniformly onX.

Corollary 2.4. Letθ ≥ 0 and letp be a real number withp > 2. Letf : X → Y be a mapping satisfying2.26 andf0 0. ThenQx : N-limn→ ∞4nfx/2nexists for each x ∈ X and defines a quadratic mappingQ:X → Ysuch that if for someδ >0, α >0

Nfxyfx−y−2fx−2fy, δθxpyp≥α 2.34

N

fx−Qx, 2δθ 2p−4x

p_≥_α _2.35

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lim t→ ∞N

fx−Qx, 8

2p−4tθx

p₁ _2.36

uniformly onX.

3. Generalized Hyers-Ulam Stability of the Quadratic

Functional Equation

1.2

In this section, we prove the generalized Hyers-Ulam stability of the quadratic functional equation1.2in fuzzy Banach spaces.

Lemma 3.1. LetV andWbe real vector spaces. If a mappingf :V → Wsatisfiesf0 0and

faxbyfax−by2a2_f_x _2b2_f_y _3.1

for allx, y∈V, then the mappingf:V → Wis quadratic, that is,

fxyfx−y2fx 2fy 3.2

for allx, y∈V.

Proof. Assume thatf:V → Wsatisfies3.1. Lettingy0 in3.1, we get

2fax 2a2_f_x _3.3

for allx∈V.

Lettingx0 in3.1, we get

fbyf−by2b2_f_y _3.4

for ally∈V. Replacingyby−yin3.4, we get

f−byfby2b2_f₋_y _3.5

for ally∈V. It follows from3.4and3.5thatf−y fyfor ally∈V. So

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for ally∈V. Thus

faxbyfax−by2a2_f_x _2b2_f_y_2f_ax _2f_by _3.7

for allx, y∈V. Replacingaxandbybyzandwin3.7, respectively, we get

fzw fz−w 2fz 2fw 3.8

for allz, w∈V, as desired.

Theorem 3.2. Letϕ:X2 → 0,∞be a function such that

ϕx,0: ∞

n0

a−2n_ϕ_an_x,₀_<_∞ _3.9

for allx∈X. Letf :X → Y be a mapping withf0 0such that

lim t→ ∞N

faxbyfax−by−2a2_f_x₋_2b2_f_y_{, tϕ}_{x, y}₁ _3.10

uniformly onX×X. ThenQx : N-limn→ ∞fanx/a2nexists for eachx ∈ X and defines a quadratic mappingQ:X → Y such that if for someδ >0, α >0

Nfaxbyfax−by−2a2_f_x₋_2b2_f_y_{, δϕ}_{x, y}_≥_α _3.11

N

fx−Qx, δ a2ϕx,0

≥α 3.12

for allx∈X.

lim t→ ∞N

fx−Qx, tϕx,01 _3.13

uniformly onX.

Proof. For a givenε >0, by3.10, we can find somet0>0 such that

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for allt≥2t0. By induction onn, we show that

N a2n_f_x₋_f_an_x_,t 2

n−1

k0

a2n−2k−2_ϕ_ak_x,₀_≥₁₋_ε _3.15

for allt≥2t0, allx∈X, and alln∈N. Lettingy0 in3.14, we get

N2fax−2a2_f_x_{, tϕ}_x,₀_≥₁₋_ε _3.16

for allx∈Xand allt≥2t0. So we get3.15forn1. Assume that3.15holds forn∈N. Then

N a2n2_f_x₋_f_an1_x_,t 2

n

k0

a2n−2k_ϕ_ak_x,₀

≥min

N a2n2_f_x₋_a2_f_an_x_{, t} 0

n−1

k0

a2n−2k_ϕ_an_x,₀

,

Na2_f_an_x₋_f_an1_x_{, t}

0ϕanx,0

≥min{1−ε,1−ε}1−ε.

3.17

This completes the induction argument. Lettingtt0and replacingnandxbypandanxin

3.15, respectively, we get

N f_aa_2nnx−fanpx

a2n2p , t0 a2n2p

p−1

k0

a2p−2k−2_ϕ_ank_x,₀_≥₁₋_ε _3.18

for all integersn≥0, p >0.

It follows from3.9and the equality

p−1

k0

a−2n−2k−2_ϕ_ank_x,₀ 1 a2

np−1

kn

a−2k_ϕ_ak_x,₀ _3.19

that for a givenδ >0 there is ann0∈Nsuch that

t0 a2

np−1

kn

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for alln≥n0andp >0. Now we deduce from3.18that

The first four terms on the right-hand side of the above inequality tend to 1 asn → ∞, and the fifth term is greater than

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for allx ∈ X. Letx ∈ X. By the same reasoning as in the beginning of the proof, one can deduce from3.18that

N a2n_f_x₋_f_an_x_{, δ}n−1 k0

a2n−2k−2_ϕ_ak_x,₀_≥_α _3.26

for all positive integersn. Lett >0. We have

Nfx−Qx, δϕnx,0 t≥min

N

fx−f_aa_2nnx, δϕnx,0

, N

_f_a_n_x

a2n −Qx, t

.

3.27

Combining3.26and3.27and the fact that lim_n_{→ ∞}Nfanx/a2n−Qx, t 1, we observe that

Nfx−Qx, δϕnx,0 t≥α 3.28

for large enoughn ∈ N. Thanks to the continuity of the functionNfx−Qx,·, we see thatNfx−Qx,δ/a2_ϕ_x,₀ _t_≥_{α. Letting}_t _→ _{0, we conclude that}

N

fx−Qx,_aδ₂ϕx,0

≥α. 3.29

To end the proof, it remains to prove the uniqueness assertion. Let T be another quadratic mapping satisfying3.1and 3.13. Fixc > 0. Given that ε > 0, by3.13forQ andT, we can find somet0>0 such that

N

fx−Qx, t 2ϕx,0

≥1−ε,

N

fx−Tx, t 2ϕx,0

≥1−ε

3.30

for allx∈Xand allt≥2t0. Fix somex∈Xand find some integern0such that

t0 ∞

kn

a−2k_ϕ_ak_x,₀_< c

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lim

t→ ∞N fx−Qx, a2

a2− |a_|ptθx p

1 3.37

uniformly onX.

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